The expectation of stopped semiamarts and regularity and closedness of amarts (Q1095493)

This paper contains stopping time characterizations of \(L^ 1\)- boundedness of semiamarts and of uniform integrability of \(L^ 1\)- bounded amarts. Reviewer's remark: The result on semiamarts asserts that a semiamart \(\{X_ n|\) \(n\in {\mathbb{N}}\}\) is \(L^ 1\)-bounded if and only if \(\int_{\{\nu <\infty \}}| X_{\nu}| dP<\infty\) holds for each stopping time \(\nu\). This result is known: In view of the lattice property of \(L^ 1\)-bounded semiamarts, the ``only if'' part is due to \textit{U. Krengel} and \textit{L. Sucheston} [Bull. Am. Math. Soc. 83, 745-747 (1977; Zbl 0336.60032), and in J. Kuelbs (ed.), Probability on Banach spaces (1978; Zbl 0394.60002), pp. 197-266]; the ``if'' part is due to the reviewer [C. R. Acad. Sci., Paris, Sér. A 287, 663-665 (1978; Zbl 0394.60045), and J. Multivariate Anal. 10, 123-134 (1980; Zbl 0418.60045)].